Anders Björn (Linköping University)
Characterizations and removability of p-superharmonic
functions on
metric spaces
Abstract.
Superharmonic functions are used as a tool to study
harmonic functions.
They are, e.g., used to define Perron solutions of
the Dirichlet (i.e. boundary value) problem for harmonic
functions.
In connection with nonlinear p-harmonic functions
on weighted R^n,
p-superharmonic functions were used
in the monograph Heinonen-Kilpeläinen-Martio [1].
An important feature of the definition is
that the bounded p-superharmonic functions
are exactly the lower semicontinuously regularized
supersolutions
(there are also unbounded p-superharmonic functions which
are not
supersolutions).
In metric spaces, Kinnunen-Martio [2], realized
that the definition from [1] was not so well suited.
They proposed a different definition, which in the
(weighted) Euclidean
case is
equivalent to the definition in [1].
In connection with quasiminimizers, they gave a third
definition in
[3],
and showed that a p-superharmonic function according to
the third
definition
is also p-superharmonic according to the second
definition.
It turns out that the second and third definitions are
equivalent and always imply the first definition,
whereas the first definition implies the other two if
there are enough
regular domains.
In this talk I will discuss this and give several other
characterizations
of superharmonicity in metric spaces.
If time permits, I will also discuss removable
singularities
for p-harmonic and p-superharmonic functions.
Relatively closed sets of capacity zero are always
removable,
but I will discuss new counterexamples of sets of positive
capacity (and even positive measure) removable for
p-harmonic
functions.
References
[1] Heinonen, J., Kilpeläinen, T. and Martio, O.,
Nonlinear Potential
Theory of Degenerate Elliptic Equations, Oxford University
Press,
Oxford, 1993.
[2] Kinnunen, J. and Martio, O., Nonlinear potential
theory on metric
spaces, Illinois Math. J. 46 (2002), 857-883.
[3] Kinnunen, J. and Martio, O., Potential theory of
quasiminimizers,
Ann. Acad. Sci. Fenn. Math. 28 (2003), 459-490.
Back to talks